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20
votes
4answers
1k views

Eigenvalues of permutations of a real matrix: can they all be real?

For a matrix $M\in GL(n,\mathbb R)$, consider the $n!$ matrices obtained by permutations of the rows (say) of $M$ and define the total spectrum $TS(M)$ as the union of all their spectra (counting ...
9
votes
1answer
1k views

Frobenius-Perron eigenvalue and eigenvector of sum of two matrices

Suppose that I have two positive matrices, $A$, and $B$, and I know their Frobenius-Perron eigenvalues ($\lambda_A$, $\lambda_B$) and eigenvectors ($v_A$, $v_B$). I'm interested in what I can say ...
8
votes
0answers
361 views

Eigenvalues of permutations of a real matrix: how complex can they be?

This is sort of complementary to this thread. I’ll repeat the definitions here: For a matrix $M\in GL(n,\mathbb R)$, consider the $n!$ matrices obtained by permutations of the rows (say) of $M$ and ...
6
votes
2answers
294 views

Does small Perron-Frobenius eigenvalue imply small entries for integral matrices?

Suppose that $M$ is an $n \times n$ matrix where each entry is a positive integer. Then $M$ is Perron-Frobenius and so has unique largest real eigenvalue $\lambda_{\textrm{PF}}$. Does an upper ...
6
votes
5answers
313 views

The maximal eigenvalue of a symmetric Toeplitz matrix

Let $0\le x\le 1$ be a real number. Denote by $A_n(x)=(a_{ij})$ the $n$ by $n$ matrix such that $a_{ij}=x^{|i-j|}$ and let $\lambda_n(x)$ be the maximal eigenvalue of $A_n(x)$. Is there any ...
6
votes
1answer
549 views

Proving that a specific kernel is positive definite

Most theoretical papers concerning kernels assume that they are given a positive definite kernel. In this question, we want to show that a specific kernel is positive definite. We are interested in ...
6
votes
0answers
146 views

Toda Flow Embeddings

What are strategies for generating the following types of pictures: Here's what's going on here. Take a toda flow in 3 variables. The equations of motion are: ...
5
votes
1answer
256 views

Anti-bidiagonal matrix with main anti-diagonal {1,2,3,…} and first sub-anti-diagonal {-1,-2,-3,…} has eigenvalues lambda={1,-2,3,-4,…}

Consider the anti-bidiagonal matrix $B_6\in\mathbb{R}^{6\times 6}$, defined along its anti-diagonals as follows $$ B_6=\begin{bmatrix} & & & & & 6\\ & & & ...
5
votes
2answers
235 views

No Exceptional Eigenvalues of Weight 1/2 Maass Forms on $\Gamma_0(4)$?

Some colleagues and I were wondering if there is a citation out there which shows there are no exceptional eigenvalues, $\lambda$, of classical weight 1/2 Maass forms on $\Gamma_0(4)$, which is to say ...
5
votes
2answers
201 views

Probabilistic Interpretation of First Dirichlet Eigenvalue?

The first Dirichlet eigenvalue of a compact domain $\Omega\subset\mathbb{R}^n$ with smooth boundary is the smallest positive number for which there exists a non-trivial solution to $$ -\Delta\psi = ...
5
votes
1answer
216 views

Separating the spectrum of a Hermitian matrix

Given Hermitian matrix $A$, I would like to perturbate it so that its eigenvalues become well-separated. Specifically, let $A$ be some Hermitian matrix, and let $G$ be a Gaussian matrix, with each ...
5
votes
0answers
70 views

Eigenvalues of Random Regular Bipartite Graphs

I am looking for a way of getting a good estimate of the eigenvalues of random bipartite d-regular graphs. The literature has very precise values the proofs of which are very involved and since I am ...
5
votes
0answers
202 views

Existence of a matrix product from its eigenvalues

Let A and B be two positive definite, real, symmetric matrices. The eigenvalues of A, B and AB, denoted by $\lambda(X)$, obey the relation (from Bhatia): $$ \lambda^\downarrow(A) \cdot ...
4
votes
2answers
219 views

Is there an easy way to tell if all eigenvalues of a unitary or self-adjoint matrix only have eigenvalues of multiplicity two?

I am interested in a class of $2n\times 2n$ unitary matrices with complex entries (if you prefer, we can replace "unitary" with "self-adjoint"). I know that all the eigenvalues of matrices in this ...
4
votes
1answer
217 views

the eigenvalues of a generalized circulant matrix

A $2k\times 2k$ circulant matrix $\ C$ takes the form \begin{align} C= \begin{bmatrix} c_0 & c_{2k-1} & \dots & c_{2} & c_{1} \\ c_{1} & c_0 & c_{2k-1} & & c_{2} ...
4
votes
3answers
223 views

Is this function well studied?

Let $A_1,\dots,A_L$ be $N\times N$ hermitian matrices. Define the simplex \begin{align} \mathcal{S}=\left\{[x_1,\dots,x_L]\mid x_i\geq 0,~\sum_{i=1}^{L}x_i=1 \right\} \end{align} and consider the ...
4
votes
1answer
213 views

Lower bounds on matrix eigenvalues

Let $A$ be a real $n\times n$ matrix and let $\mu_1, \dots, \mu_n$ the (generalized, complex) eigenvalues of $A$. Assume that $$ 0 < \alpha < \mathrm{Re}(\mu_1) < \dots < ...
4
votes
1answer
147 views

Epidemic threshold

Need some help / ideas to proceed. Stuck for a while on this. In the literature of epidemic theory, it is found that the epidemic threshold is $1/\lambda_{max}(A)$ where $\lambda_{max}(A)$ is the ...
4
votes
2answers
982 views

rank-one perturbation of a matrix corresponding to a specific spectrum

Let $A$ be a real symmetric matrix whose spectrum is $\lambda_1,\lambda_2,\ldots,\lambda_n$. Let $A'$ be the matrix obtained by adding a perturbation to $A$. The requirement is that only the second ...
4
votes
1answer
223 views

Question about a (relatively simple looking) differential operator and its eigenvalues

A colleague and I are interested in a specific differential operator on the reals. The differential operator L is of the form $L=-(1+x^{2})\frac{d^{2}}{dx^{2}}+c_{1}x\frac{d}{dx}+c_{2}x^{2}$ for ...
4
votes
1answer
227 views

A complex sequence with positive values

Let $\lambda_1,\dots,\lambda_d$ be complex numbers that constitute the spectrum of a nonnegative integer matrix, and $P_1,\dots, P_d$ be complex polynoms, such that the sequence $$u_n=\Sigma_{i=1}^d ...
4
votes
2answers
1k views

Eigenvalues of a Symmetric Positive Semi-Definite (PSD) matrix after rank one update

I have a Symmetric Positive Semi-Definite matrix $A$ which i know its eigenvalue and eigenvectors. let $v$ and $u$ be a random column vector. i want to know if it is possible to have eigenvalues of ...
3
votes
2answers
356 views

Prove log of eigenvalues are dense in R?

Suppose you have the set of all possible $n$ x $n$ square adjacency matrices where $n$={1,2,3,4...}. For each matrix, compute the logarithm of the largest eigenvalue. Is it true that the set of ...
3
votes
2answers
1k views

Upper bounds on eigenvalues of PSD matrix?

Suppose A is a symmetric positive semidefinite matrix. Is there a way to upper bound the largest eigenvalue using properties of its row sums or column sums? For instance, the Perron–Frobenius ...
3
votes
1answer
515 views

Why are 1 and -1 eigenvalues of this matrix?

This is a subject I've been working on for a very long time now, but still did not manage to fully understand the interesting properties of this matrix $\mathbf{A}$. First, let's define two matrices: ...
3
votes
1answer
290 views

Matrix norms / eigenvalues / singular values / another thing

OK, here is what is probably a stupid question. Let $M$ be a non-symmetric real matrix: for example, the shear matrix $\left( \begin{array}{cc} 1 & 1 \\\ 0 & 1 \end{array} \right)$. There ...
3
votes
2answers
473 views

What is known about the spectrum of a Cauchy matrix?

Math people: A Cauchy matrix is an $m$-by-$n$ matrix $A$ whose elements have the form $a_{i,j} = \frac{1}{x_i-y_j}$, with $x_i \neq y_j$ for all $(i, j)$, and the $x_i$'s and $y_i$'s belong to a ...
3
votes
1answer
765 views

Connection between eigenvalues of matrix and its Laplacian.

Hello! There are two definitions of graph spectrum: 1) Eigenvalues of adjacency matrix $A$. 2) Eigenvalues of Laplacian of adjacency matrix ($L$). Different sources offer different properties based ...
3
votes
1answer
89 views

Eigenfunctions to 2nd-order Differential Operators: Relation between Frobenius Series Solution and Eigenfunction Normalised to the Delta Function

Consider the 2nd-order linear ODE $x f^{''}(x) + x (\beta - 2 \alpha x) \kappa / \sigma f^{'}(x) - 1 / \sigma \left[ 2 \alpha \kappa - \lambda^2 (\beta - 2 \alpha x)^2 \right] f(x) = 0$, where ...
3
votes
1answer
197 views

When is this matrix singular?

Consider matrix $A$ with $(j,k)$′th entry $A_{j,k}=\sin(\omega_j t_k+\phi_j),\,\forall j,k\in\{1,2,...,n\}$, where $\omega_j,t_k,\phi_j\in \mathbf R$. 1) For $t_k=k$, what is the condition on ...
3
votes
2answers
299 views

Eigenvectors and eigenvalues of Tridiagonal matrix with varying diagonal elements

is it possible to analytically evaluate the eigenvectors and the eigenvalues of a tridiagonal $n\times n$ matrix of the form : \begin{pmatrix} 1 & b & 0 & ... & 0 \\\ b & 2 ...
3
votes
1answer
465 views

Condition for block symmetric real matrix eigenvalues to be real

I have a (2nx2n) block symmetric matrix that in the simplest case (n=2) looks like: $$ M_2 = \begin{bmatrix} a_1 & 0 & b_{1,2} & -b_{1,2}\\\ 0 & -a_1 & b_{1,2} & -b_{1,2}\\\ ...
3
votes
0answers
85 views

Bound on the ratio of top 2 eigenvalues

Let $P$ be a $(n+1) \times (n+1)$ stochastic matrix such that $P_{ij}=\tau$ if $i \neq j$ and $P_{ii} = (1 - n\tau)$ where $0<\tau < \frac{1}{n+1}$. It is clear that the largest eigenvalue of ...
3
votes
0answers
82 views

Classes for which the Spectrum determines a Convex Shape

Given a planar domain $\Omega \subset \Bbb{R}^2$ bounded and open we can associate to it the spectrum of the Laplace operator with Dirichlet boundary condition. It is known that there are planar ...
3
votes
0answers
146 views

Eigenvalues vs.matrix sparsity

For an n X n matrix whose entries are constrained to be in some [x,y], is the maximum absolute eigenvalue of the matrix a function of its sparsity? Is there a closed-form expression that states this ...
2
votes
1answer
868 views

Non symmetric matrices with real eigenvalues

Consider the following block matrix $A=\pmatrix{A_1 & A_2\cr kA_2^\top & A_3}$ where $A_1$ is a symmetric matrix, $A_3$ is diagonal matrix and all entries of $A$ are real and non-negative. ...
2
votes
1answer
105 views

the impossibility of exactly computing eigenvalues [closed]

I is well known that there is no explicit formula for the eigenvalues of a general matrix (see e.g. Wikipedia). This result is a consequence of (1) Abel's theorem, stating that there is no explicit ...
2
votes
2answers
158 views

Matrices with real spectrum

Assume you have a non-symmetric real square matrix of all whose eigenvalues are real. Can anything be said about it? Is it unitarily equivalent to a symmetric matrix? EDIT: Is it at least similar to ...
2
votes
3answers
480 views

Analytic perturbation of the eigenvalues/eigenvectors of non-Hermitian matrix

Consider a matrix function $A(x)$, analytically depending on single parameter $x$. Consider the eigenvalue/eigenvector pair of $A(0)$, namely $\lambda(0)$ and $w(0)$. The question is whether we can ...
2
votes
1answer
514 views

Trace inequality for matrices with determinant 1

Let $A$ and $B$ be two matrices with $\det(A)=\det(B)=1$. Does it follow that $\sqrt{\mathrm{tr}(A^TB^TBA-I)}\le\sqrt{\mathrm{tr}(A^TA-I)}+\sqrt{\mathrm{tr}(B^TB-I)}$ I suspect that this can be ...
2
votes
1answer
91 views

Proof of eigenvalue stability inequality via Courant-Fischer min-max theorem

Dr. Tao in his notes on eigenvalue inequalities uses Courant-Fischer min-max theorem to prove the eigenvalue stability inequality. Specifically, I am looking for proof of Eq. (13) where Dr. Tao ...
2
votes
1answer
77 views

Maximising a Rayleigh quotient over a subspace

Let $M\in\mathbb{R}^{n\times n}$ be symmetric positive definite and consider a matrix $Q\in\mathbb{R}^{n\times m}$ ($m<n$) with orthonormal columns ($Q^TQ=I$). I'm interested in finding an exact ...
2
votes
3answers
331 views

LU decomposition

Consider a $N \times N$ symmetric real matrix $A$: $A_{ij} = (\sum_{k=1}^N n_{ik}) \delta_{ij} - n_{ij}$, where $n_{ij}$ is a real symmetric matrix whose elements are equal to $1$ or $0$. $A$ has one ...
2
votes
1answer
444 views

Gaussian kernel eigenfunctions

I'm a newbie and may be this question is bit simple for you but pardon me if it's too simple and provide me some references. What is the eigenfunction of a multivariate Gaussian kernel: ...
2
votes
1answer
228 views

quadratic programming on hypercube

I want to maximize a quadratic form $\mathbf x^T\mathbf Q\mathbf x$ and also want to find out which vector $\mathbf x$ maximizes the quadratic form when $\mathbf Q$ is an $n\times n$ positive ...
2
votes
1answer
204 views

When is there a solution to these coupled eigenvalue equations?

I am trying to find the fixed point of a dynamical system, which requires solving two coupled eigenvalue-like equations. These equations are, in general, overconstrained. I'd like to have a simple ...
2
votes
3answers
312 views

eigenvalue of Laplacian matrix

If we have a Laplacian matrix $\boldsymbol{A}$ such that \begin{align} &A_{ii} >0 \\ &A_{ii}=-\sum_{j\neq i}A_{ij} \end{align} with known eigenvalues $\lambda_i$. Define the matrix ...
2
votes
2answers
157 views

Is my use of the eigendecomposition correct here?

I'm exploring different techniques to efficiently solve some matrix equations. My situation is that I have a matrix $\textbf{H} = \textbf{J}^T \textbf{J}$, where $\textbf{J}$ is a matrix with no ...
2
votes
1answer
194 views

Multiple eigenvalues over imperfect fields

Let $K$ be a field. For a matrix $A\in GL_n(K)$ we can find the Jordan normal form $A'$ in $GL_n(\overline{K})$, where $\overline{K}$ is the algebraic closure of $K$. We write $j_\alpha(A)$ for the ...
2
votes
1answer
64 views

Is there a general way to determine the Laplacian of the eigenvalues of a real symmetric matrix? [closed]

I have a real symmetric $3\times3$ matrix $\mathbf{M}(\mathbf{r}$) which depends on $\mathbf{r} \in \mathbb{R}^3$. Each eigenvalue can be considered a scalar field $e_i(\mathbf{r})$ over ...